# Total Internal Reflection

Technology and Art

# Pure Mathematics

• ## Functional Analysis Exercises 12 : Linear Operators

13 December 2021

This post lists solutions to the exercises in the Linear Operators section 2.6 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics, Kreyszig
• ## Functional Analysis Exercises 11 : Compactness and Finite Dimension

2 December 2021

This post lists solutions to the exercises in the Compactness and Finite Dimension section 2.5 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics, Kreyszig
• ## Functional Analysis Exercises 10 : Finite Dimensional Normed Spaces and Subspaces

22 November 2021

This post lists solutions to the exercises in the Finite Dimensional Normed Spaces and Subspaces section 2.4 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics, Kreyszig
• ## Functional Analysis Exercises 9 : Further Properties of Normed Spaces

19 November 2021

This post lists solutions to the exercises in the Further Properties of Normed Spaces section 2.3 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics, Kreyszig
• ## Functional Analysis Exercises 8 : Normed and Banach Spaces

11 November 2021

This post lists solutions to the exercises in the Normed Space, Banach Space section 2.2 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics, Kreyszig
• ## Functional Analysis Exercises 7 : Vector Spaces

3 November 2021

This post lists solutions to the exercises in the Vector Space section 2.1 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics, Kreyszig
• ## Functional Analysis Exercises 6 : Completion of Metric Spaces

25 October 2021

This post lists solutions to the exercises in the Completion of Metric Spaces section 1.6 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics, Kreyszig
• ## Functional Analysis Exercises 5 : Completeness Proofs

16 October 2021

This post lists solutions to the exercises in the Completeness Proofs section 1.5 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics, Kreyszig
• ## Functional Analysis Exercises 4 : Convergence, Cauchy Sequences, and Completeness

12 October 2021

This post lists solutions to the exercises in the Convergence, Cauchy Sequences, and Completeness section 1.4 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics, Kreyszig
• ## Assorted Analysis Proofs

11 October 2021

This post lists assorted proofs from Analysis, without any particular theme.

Tags: Mathematics, Proof, Analysis, Pure Mathematics
• ## Functional Analysis Exercises 3 : Sets, Continuous Mappings, and Separability

7 October 2021

This post lists solutions to many of the exercises in the Open Set, Closed Set, Neighbourhood section 1.3 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics, Kreyszig
• ## General Proof Tactics for Real and Functional Analysis

29 September 2021

This article represents a (very short) collection of my ongoing notes on proof tactics I’ve found useful when I’ve been stuck trying to solve proof exercises. I aim to continue documenting these in as much detail as possible. These are mostly aids while building intuition about how to prove something, and gradually should become part of one’s mental lexicon.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics
• ## Functional Analysis Exercises 2 : Distance Metrics

28 September 2021

This post lists solutions to many of the exercises in the Distance Metrics section 1.2 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is a work in progress, and proofs may be refined over time.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics, Kreyszig
• ## Important Inequalities in Functional Analysis

27 September 2021

Continuing my self-study of Functional Analysis, this post describes proofs for the following important inequalities in the subject:

• Young’s Inequality
• Hölder’s Inequality
• Minkowski’s Inequality
Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics
• ## Functional Analysis Exercises 1 : Distance Metrics

21 September 2021

This post lists solutions to many of the exercises in the Distance Metrics section 1.1 of Erwin Kreyszig’s Introductory Functional Analysis with Applications. This is definitely a work in progress, and proofs may be refined or added over time.

Tags: Mathematics, Proof, Functional Analysis, Pure Mathematics, Kreyszig
• ## Functional Analysis: Norms, Operators, and Some Theorems

19 July 2021

This article expands the groundwork laid in Kernel Functions: Functional Analysis and Linear Algebra Preliminaries to discuss some more properties and proofs for some of the properties of functions that we will use in future discussions on Kernel Methods in Machine Learning, including (but not restricted to) the construction of Reproducing Kernel Hilbert Spaces.

Tags: Mathematics, Theory, Operator Theory, Functional Analysis, Pure Mathematics
• ## Functional and Real Analysis Notes

18 July 2021

These are personal study notes, brief or expanded, complete or incomplete. Some concepts here will be alluded to in full-fledged Machine Learning posts.

Tags: Mathematics, Theory, Notes, Functional Analysis, Pure Mathematics
• ## Real Analysis: Patterns for Proving Irrationality of Square Roots

9 July 2021

ld Continuing on my journey through Real Analysis, we will focus here on common proof patterns which apply to irrational square roots. These patterns apply to the following sort of proof exercises:

Tags: Real Analysis, Mathematics, Proof, Pure Mathematics
• ## Real Analysis Proofs #1

18 May 2021

Since I’m currently self-studying Real Analysis, I’ll be listing down proofs I either initially had trouble understanding, or enjoyed proving, here. These are very mathematical posts, and are for personal documentation, mostly.

Tags: Real Analysis, Mathematics, Proof, Pure Mathematics